Continuous random variables

So far, we've looked at discrete outcomes in the sample space where we could find the probability of a certain outcome. But now, in the continuous space, we will find the probability of our outcome being in a particular interval or range.

Now, to find the distribution of X, we need to define a function, f, so that the probability of X must lie in the interval 

Formally, a random variable, , is continuous if, in a function,  so that we have the following:

We call the function, f, the probability density function (PDF) and it must satisfy the following:

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There is another distribution function that is important for us to know, known as the cumulative distribution function. If we have a random variable, X, that could be continuous or discrete, then, , where F(x) is increasing so that x→∞ and F(x)→1.

When dealing with continuous random variables such as the following, we know that F is both continuous and differentiable:

So, when F is differentiable, then F'(x) = f(x).

An important fact to note is that .

This leads us to the concept of uniform distribution, which, in general, has the following PDF:

So, we have the following:

This is the case for .

We write  if X follows a uniform distribution on the [a, b] interval.

Now, let's suppose our random variable is an exponential random variable and has the added λ parameter. Then, its PDF is  and  for all .

We write this as , so we have the following:

It is also very important to note that the exponential random variable, such as the geometric random variable, is memory-less; that is, the past gives us no information about the future.

Just as in the discrete case, we can define the expectation and variance in the case of continuous random variables. 

The expectation for a continuous random variable is defined as follows:

But, say . Then, we have the following:

In the case of continuous random variables, the variance is defined as follows:

This gives us the following:

Now, for example, let's take . We can find the expected value of X as follows:

Its variance can be found as follows:

Now, we have a good handle of expectation and variance in continuous distribution. Let's get acquainted with two additional terms that apply to PDFs—mode and median.

The mode in a PDF is the value that appears the most; however, it is also possible for the mode to appear more than once. For example, in a uniform distribution, all the x values can be considered as the mode.

Say we have a PDF, f(x). Then, we denote the mode as , so  for all cases of x.

We define the median as follows:

However, in a discrete case, the median is as follows:

Many times, in probability, we take the sample mean instead of the mean. Suppose we have a distribution that contains all the values that X can take. From it, we randomly sample n values and average it so that we have the following: